Mathematics (Project Maths – Phase...

2014. S234

Coimisiún na Scrúduithe Stáit

State Examinations Commission

Junior Certificate Examination 2014

Mathematics (Project Maths – Phase 2)

Paper 1

Higher Level

Friday 6 June – Afternoon, 2:00 to 4:30

300 marks

Examination number

Centre stamp

Running total

For examiner

Question Mark Question Mark

1 11

2 12

3 13

4 14

5

6

7

8

9

10 Total

Grade

Junior Certificate 2014 Page 2 of 19 Project Maths, Phase 2 Paper 1 – Higher Level

Instructions

There are 14 questions on this examination paper. Answer all questions.

Questions do not necessarily carry equal marks. To help you manage your time during this examination, a maximum time for each question is suggested. If you remain within these times you should have about 10 minutes left to review your work.

Question 14 carries a total of 50 marks.

Write your answers in the spaces provided in this booklet. You may lose marks if you do not do so. There is space for extra work at the back of the booklet. You may also ask the superintendent for more paper. Label any extra work clearly with the question number and part.

The superintendent will give you a copy of the Formulae and Tables booklet. You must return it at the end of the examination. You are not allowed to bring your own copy into the examination.

You will lose marks if all necessary work is not clearly shown.

Answers should include the appropriate units of measurement, where relevant.

Answers should be given in simplest form, where relevant.

Write the make and model of your calculator(s) here:


Question 1 (Suggested maximum time: 5 minutes)

(a) Place the following numbers in order, starting with the smallest:

3

2 1·4 2

(b) Which one of the following is not a rational number? Explain your answer.

1

37

3·142 22

7 π

(c) (i) Find the values of 24 1

,13

n + where { }17, 19, 21 .n∈

n 13

14 2 +n

17

19

21

(ii) State which one of your answers is a natural number, and explain your choice.

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Junior Certif

Question 2

(a) John He sa He a

(i)

(ii)

(b) The Sthe evalue

Expl

Re

Re

ficate 2014

2

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Complete

Give two r

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ain why it d

eason 1:

eason 2:

he has a mhe uses the fat these form

the table.

p

0

1

2

3

4

5

reasons why

hematician a2 41n n− +

than 41.

does not giv

Pag

ethod for fiformulas in mulas will g

6p +

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and physicisgives a prim

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ge 4 of 19

nding all prthe table be

generate onl

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od is not full

st, Euler, firme number

number for

(Sugge

rime numbeelow, he wily the prime

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ly correct.

rst noticed (for all posi

41.n =

ested maxim

ers. ll generate t

e numbers.

+ 5

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(in 1772) thaitive integer

Project MPaper 1 –

mum time:

the prime n

at r

Maths, Phase 2Higher Level

5 minutes)

numbers.

2 l

)


A B

C


(a) The sets A, B, and C are as follows:

A = {2, 3, 4, 5, 6}, B = {2, 4, 6, 8, 10}, and C = {1, 4, 8, 12, 14}.

(i) Complete the Venn diagram.

(ii) List the elements of each of the following sets:

A B∩ = _________________________ ( )\B A C∩ = _________________________

( ) ( )\ \B A B C∪ = __________________________

(iii) Write down a null set, in terms of A, B, and C. _______________________________ (b) In a table quiz, 100 questions were asked. Team M answered 72 questions correctly.

Team N answered 38 questions correctly.

(i) Find, with the aid of the Venn diagram, the minimum number of questions answered correctly by both teams.

(ii) Find, with the aid of the Venn diagram, the maximum number of questions answered

correctly by both teams.

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Minimum =

U

M N

Maximum =

U

M N



(a) Factorise fully 29 6 12 8 .a ab ac bc− + −

(b) Factorise 2 29 16 .x y−

(c) Use factors to simplify the following: 62

422

2

−++xx

xx.


Solve the following inequality and show the solution on the number line.

17 1 3 13, x x− ≤ − < ∈

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7



Below are three containers, labelled 1, 2, and 3. Water is poured into each container at a constant rate, until it is full.

1 2 3

The three graphs, A, B, and C, show the height of the water, h, in the containers after time t.

(a) Write A, B, and C in the table below to match each container to its corresponding graph.

(b) Another container is shown below. Water is also poured into this container at a constant rate

until it is full. Sketch the graph you would expect to get when plotting height (h) against time (t) for this container.

Container 1 2 3

Graph

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A

h

t

h

t

h

t

B C

h

t



Last year Elena had a gross income of €36 960. She had to pay Universal Social Charge (USC) and income tax on her gross income. The rates and bands of USC are as follows.

(i) Find the amount of USC that was deducted from Elena’s gross income last year.

(ii) The standard rate of income tax was 20% and the higher rate was 41%. The standard rate cut-off point was €32 800. Elena paid a total of €4965·60 income tax last year.

Find Elena’s tax credits for the year.

Income band Rate of USC

Up to €10 036 2%

Between €10 036 and €16 016 4%

Above €16 016 7%


(iii) Find Elena’s total deduction (USC and income tax) as a percentage of her gross income. Give your answer correct to the nearest percent.


The table shows the height, in metres, of a ball at various times after being kicked into the air.

(i) Is the pattern of heights in the table linear, quadratic, or exponential? Explain your answer.

Time (seconds) 0 0⋅5 1 1⋅5 2 2⋅5 3

Height (metres) 0⋅3 3⋅4 5⋅7 7⋅2 7⋅9 7⋅8 6⋅9

(ii) Estimate the height of the ball after 3⋅5 seconds.

(iii) Estimate the total time the ball spends in the air. Justify your answer.

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Jack and Sarah are going on a school tour to England. They investigate how much different amounts of sterling (£) will cost them in euro (€). They each go to a different bank.

Their results are shown in the table below.

(i) On the grid below, draw graphs to show how much the sterling will cost Jack and Sarah, for up to £80.

Amount of sterling (£)

Cost in euro (€) for Jack

Cost in euro (€) for Sarah

20 33 24

40 56 48

60 79 72

80 102 96

Amount of Sterling (£)

Am

ount

of

Eur

o (€

)

20 40 60 80

10

20

30

40

50

60

70

80

90

100


(ii) Using the table, or your graph, find the slope (rate of change) of Jack’s graph. Explain what this value means. Refer to both euro and sterling in your explanation.

(iii) Write down a formula to represent what Jack must pay, in euro, for any given amount of

sterling. State clearly the meaning of any letters you use in your formula.

(iv) Write down a formula to represent what Sarah must pay, in euro, for any given amount of sterling. State clearly the meaning of any letters you use in your formula.

(v) Using your formulas from (iii) and (iv), or otherwise, find the amount of sterling Jack and

Sarah could buy that would cost them the same amount each in euro.

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Slope:

Explanation:



(a) The graphs of the functions 2( ) 2 3f x x x= + − and 2( ) 2 3g x x x= − − + are shown below. Identify each graph by writing ( )f x or ( )g x in the space provided below the graph.

Answer: _______ Answer: _______

(b) The graphs of the functions ( )y h x= and ( )y k x= are shown below.

( )h x ( )k x

Write down the roots of each function. Hence, or otherwise, write down an equation for each function.

-4 -3 -2 -1 1 2 3

-6

-4

-2

2

4

6

8y

x

-3 -2 -1 1 2 3 4

-6

-4

-2

2

4

6

8 y

x

-4 -3 -2 -1 1 2

-6

-4

-2

2

4 y

x

-4 -3 -2 -1 1 2

-4

-2

2

4

6 y

x

Equation: ( )h x =

Equation: ( )k x =

Roots of ( ) :h x

Roots of ( ) :k x



x is a real number. One new number is formed by increasing x by 1. A second new number is formed by decreasing x by 2.

(i) Write down each of these new numbers, in terms of x. (ii) The product of these two new numbers is 1. Use this information to write an equation in x. (iii) Solve this equation to find the two possible values of x.

Give each of your answers correct to 3 decimal places.


(a) Simplify ( )( )6 3 2 1 .x x− −

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Increase x by 1:

Decrease x by 2:


(b) Simplify ( ) ( )3 23 2 3 2 1 .x x x x− − + ÷ −

(c) (i) Solve the simultaneous equations:

2 3 18

5 9 10.

x y

x y

− =+ = −

(ii) Verify your answer to (c)(i).



(i) Use the diagram on the right to calculate the value of x. Give your answer in surd form. (ii) Use the diagram below to calculate the value of y. Give your answer in surd form. (iii) A rectangle with sides of length x and y is drawn using the values of x and y from

parts (i) and (ii), as shown below.

Write the perimeter of this rectangle in the form 2a , where .a∈

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3

x 3

1

y

3

y

x



(i) g is the function : 1g x x − , where x ∈ . Find the value of each of the following.

(ii) f is the function 2: 2 6f x x x− − , where x∈.

Using the same axes and scales, draw the graphs of the functions ( )y f x= and ( )y g x= in the domain 2 3x− ≤ ≤ .

(3)g

( 2)g −


Use your graphs from (ii) to estimate:

(iii) the minimum value of ( )f x (iv) the range of values of x for which ( ) 0f x < (v) the range of values of x for which ( ) 0.g x ≥ Page running


You may use this page for extra work.


You may use this page for extra work.

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Junior Certificate 2014 – Higher Level

Mathematics (Project Maths – Phase 2) – Paper 1 Friday 6 June Afternoon, 2:00 to 4:30

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